Dr. Hackney STA Solutions pg 157

Dr. Hackney STA Solutions pg 157 - Second Edition 9-15...

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Second Edition 9-15 where c = σ 2 2 . The derivative (with respect to c ) of this last expression is bf n - 1 ( bc ) - af n - 1 ( ac ), and hence is equal to zero if both c = 1 (so the interval is unbiased) and bf n - 1 ( b ) = af n - 1 ( a ). From the form of the chi squared pdf, this latter condition is equivalent to f n +1 ( b ) = f n +1 ( a ). d. By construction, the interval will be 1 - α equal-tailed. 9.53 a. E [ b length( C ) - I C ( μ )] = 2 cσb - P ( | Z | ≤ c ), where Z n(0 , 1). b. d dc [2 cσb - P ( | Z | ≤ c )] = 2 σb - 2 1 2 π e - c 2 / 2 . c. If bσ > 1 / 2 π the derivative is always positive since e - c 2 / 2 < 1. 9.55 E[ L (( μ,σ ) , C )] = E [ L (( μ,σ ) , C ) | S < K ] P ( S < K ) + E [ L (( μ,σ ) , C ) | S > K ] P ( S > K ) = E L (( μ,σ ) , C ) | S < K P ( S < K ) + E [ L (( μ,σ ) , C ) | S > K ] P ( S > K ) = R L (( μ,σ ) , C ) + E [ L (( μ,σ ) , C ) | S > K ] P ( S > K ) , where the last equality follows because C = if S > K . The conditional expectation in the second term is bounded by E [ L (( μ,σ ) , C ) | S > K ] = E [ b length( C ) - I C ( μ ) | S > K ] = E [2 bcS - I C ( μ ) | S > K ] > E [2 bcK - 1 | S > K ] (since S > K and I C 1) = 2 bcK - 1 , which is positive if K >
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